One easily sees that in inner product spaces BJ orthogonality is equivalent to the usual one. More importantly, several key properties of orthogonality in inner product spaces are inherited within BJ orthogonality. For example, among all the vectors from a subspace \(Y\subseteq X\)  a vector \(y_0\in Y\)  is the best approximation to \(x\)  (in the sense that \(\Vert x-y_0\Vert =\min _{y\in Y}\Vert x-y\Vert \) )
precisely when \(x-y_0\)  is BJ orthogonal to \(Y\) . Another trivial implication, valid for every BJ orthogonal (i.e., \(\mathcal {M}\perp \mathcal {N}\) )
closed
subspaces \(\mathcal {M}, \, \mathcal {N}\)  in a Banach space \(X\)  is that they have trivial intersection (excuse the pun) and their sum, \(\mathcal {M}+\mathcal {N}\)  is also a closed subspace — this can be seen by noting that \(\mathcal {M}\perp \mathcal {N}\)  implies that the projection from \(\mathcal {M}+\mathcal {N}\)  onto \(\mathcal {M}\) 
is norm-one.
In this respect, we mention here a classical result by Anderson [1]} that the range of an inner-derivation, induced by a normal operator \(N\)  on a complex Hilbert space, is always BJ orthogonal to its kernel (this result has been greatly extended in various directions [2]}).
Among other applications of BJ orthogonality we mention Bhatia–Šemrl's alternative proof that the diameter of the unitary orbit of a given complex matrix \(A\)  equals \(2d(A,\mathbb {C}I)\)  (see [3]}).
